The Pendulum, Finally Explained

Class 6 grouped circular motion and oscillatory motion together under one label, periodic motion, since both repeat after a fixed interval. This chapter draws a sharper line between them. A motion is periodic if it repeats itself at regular intervals of time; the smallest such interval is its period, T, and the number of repetitions per second is its frequency, ν = 1/T, measured in hertz (Hz), where 1 Hz means exactly one repetition per second. A planet orbiting the Sun is periodic, but it never doubles back, it simply keeps going around, so it is periodic without being oscillatory. Oscillatory motion is the stricter case: periodic motion that is also to-and-fro about some fixed mean position, like a pendulum, a plucked guitar string, or a mass bobbing on a spring. Every oscillatory motion is periodic, but as the orbiting planet shows, not every periodic motion is oscillatory, a distinction Class 6's simpler treatment did not need to draw, but which matters once you start describing oscillation mathematically. A body undergoing oscillation almost always has a natural equilibrium position where, left alone, it would simply stay forever; displace it slightly, and a restoring force or torque appears, pulling or pushing it back toward that equilibrium, which is exactly what sets an oscillation going in the first place.

A block on a spring, pulled to one side and released, oscillates back and forth about its equilibrium position, and its displacement x, measured from that equilibrium, turns out (for an ideal spring) to follow a strikingly clean mathematical pattern: x(t) = A cos(ωt + φ), the defining equation of simple harmonic motion (SHM). Each symbol carries a precise meaning. A, the amplitude, is the maximum displacement reached on either side, always a positive number. ω, the angular frequency, sets how rapidly the motion cycles, related to the ordinary period by ω = 2π/T; a larger ω means a faster oscillation squeezed into a shorter period. The whole quantity (ωt + φ) is called the phase, describing exactly where in its cycle the oscillation currently is, and φ itself, the phase constant, is simply the phase at t = 0, fixed by wherever the motion happened to start. Two oscillations can share the same A and ω but differ in φ, meaning they trace out the identical up-and-down pattern, just offset in time from each other, like two identical pendulums released at slightly different moments.

SHM has a genuinely elegant geometric origin. Watch a particle moving at constant angular speed ω around a circle of radius A, edge-on, so that only its shadow, its projection onto one diameter, is visible; that shadow moves back and forth, and it turns out to follow exactly x(t) = A cos(ωt + φ), the SHM equation, with φ simply the particle's starting angle on the circle. This is not a coincidence or a loose analogy, it is an exact mathematical identity: any uniform circular motion, viewed edge-on along a diameter, is indistinguishable from SHM. This single idea instantly connects two chapters: the centripetal acceleration derived for circular motion in the previous chapter, always pointing toward the centre with magnitude ω²A, projects onto the same diameter as -ω²x(t), exactly matching SHM's own acceleration formula, derived independently below. Despite this deep mathematical kinship, it is worth being precise about what is and is not shared: the actual physical force driving a swinging pendulum bears no resemblance to the centripetal force driving real circular motion; only the mathematical pattern of the resulting displacement is identical.

Differentiating x(t) = A cos(ωt + φ) once gives velocity, v(t) = -ωA sin(ωt + φ), and differentiating again gives acceleration, a(t) = -ω²A cos(ωt + φ), which is simply -ω²x(t): acceleration in SHM is always directly proportional to displacement, and always points the opposite way, back toward equilibrium. By Newton's second law, this fixes the force responsible: F = ma = -mω²x(t), which is usually written F = -kx, where k = mω² is called the force constant, exactly Hooke's law for an ideal spring, with the minus sign confirming the force always opposes displacement, restoring the system toward equilibrium rather than driving it further away. This gives two entirely equivalent ways to define SHM: by its displacement equation, x(t) = A cos(ωt + φ), or by its force law, F = -kx, and either one implies the other through differentiation and integration. A block of mass 0.4 kg attached to a spring of force constant k = 100 N/m, displaced 5 cm and released, oscillates with ω = √(k/m) = √(100/0.4) = 15.8 rad/s, a period of T = 2π/ω ≈ 0.40 s, and a maximum acceleration, at the extremes of its swing, of ω²A = (15.8)² x 0.05, about 12.5 m/s², roughly 1.3 times the acceleration due to gravity.

As a block on a spring oscillates, its kinetic energy, ½mv², and potential energy, ½kx², continuously trade places, but their sum stays exactly constant. At the extremes of the swing (x = ±A), velocity is momentarily zero, so all the energy is potential: E = ½kA². At the equilibrium position (x = 0), all the stored spring energy has been released, so all the energy is kinetic, and since energy is conserved, this kinetic energy at the centre must equal the same ½kA² found at the extremes; from this, the maximum speed follows directly as vₘₐₓ = ωA, exactly matching the coefficient in the velocity equation. In between these two extremes, the two forms of energy continuously exchange, kinetic converting to potential as the block moves away from centre, then potential converting back to kinetic as it swings back through, forever, in an ideal, friction-free system. For the spring from the previous section (k = 100 N/m, A = 5 cm), the total mechanical energy is ½ x 100 x (0.05)², or 0.125 J, present at every instant of the motion, however that 0.125 J happens to be split between kinetic and potential energy at that particular moment.

Class 7 discovered, by careful timing, that a pendulum's time period depends only on its length, not the mass of its bob, without being told why. The answer follows from treating the pendulum properly: a bob of mass m on a string of length L, displaced to angle θ from vertical, has a restoring torque about the support point of τ = -mgL sin θ, the tangential component of gravity, pulling it back toward θ = 0. Newton's law for rotation gives τ = Iα, and for a simple pendulum, treated as a point mass, the moment of inertia is I = mL², so the angular acceleration works out to α = -(g/L) sin θ. For small angles (below roughly 20°), sin θ is very well approximated by θ itself (measured in radians), simplifying this to α = -(g/L)θ, which has exactly the same mathematical form as SHM's own a(t) = -ω²x(t), confirming that a pendulum's small-angle swing genuinely is simple harmonic motion, with ω² playing the role of g/L. This immediately gives the period: T = 2π√(L/g). Notice what happened to the mass: it appeared in both the torque (mgL sin θ) and the moment of inertia (mL²), and cancelled out completely, which is precisely why Class 7's careful pendulum measurements found no dependence on the bob's mass at all, not an experimental coincidence, but a direct mathematical consequence of how mass enters the equation twice, in exactly matching ways. A pendulum of length 1 m has a period of 2π√(1/9.8), about 2.0 s; lengthen it to 4 m, four times the length, and the period becomes 2π√(4/9.8), about 4.0 s, exactly double, not four times, since period depends on the square root of length, not length directly.